#27092523

order	12032
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₄₇×#3584 ⋊ C₂
Sylow	C₄₇ ⋊ #61106

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₄₇×#3584	C₂
socle	C₂×C₄₇	C₂×D₄²
hypercenter	#3584	D₄₇
hypocenter	C₄₇	#61106
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3584

2656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 1504

A minimal faithful permutation representation — 5 generators on 79 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(49,58)(55,72)(56,73)(57,75)(59,60)(61,76)(70,71)(74,77);(48,51)(49,57)(50,63)(52,69)(53,66)(54,67)(55,71)(56,59)(58,75)(60,73)(61,77)(62,68)(64,79)(65,78)(70,72)(74,76);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,51)(49,56)(50,62)(52,66)(53,69)(54,67)(55,70)(57,59)(58,73)(60,75)(61,77)(63,68)(64,78)(65,79)(71,72)(74,76);(48,50)(49,55)(51,62)(52,68)(53,64)(54,65)(56,70)(57,74)(58,72)(59,76)(60,61)(63,66)(67,79)(69,78)(71,73)(75,77);(48,49)(50,61)(51,57)(52,60)(53,59)(54,58)(55,64)(56,66)(62,70)(63,77)(65,76)(67,75)(68,72)(69,73)(71,79)(74,78)

Polycyclic presentation:

2^8,47|f^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gh,d^b=df,f^b=fh,g^c=gh,i^c=i^46,f^d=fh

Conjugacy classes & character table

1064 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12032, 1045813);